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Counting Central Configurations at the Bifurcation Points

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Abstract

Enumeration problems for the central configurations of the Newtonian \(n\) body problem are hard for \(n>3\) in \(\mathbb{R}^{2}\) and \(n>4\) in \(\mathbb{R}^{3}\). These are problems in finding the numbers of classes of central configurations for all the masses in a parameter space of positive dimensions. Many results are obtained generically. That is, rigorous proofs of the counting problems only exists for parameters not at the bifurcation points. For the bifurcation points, only numerical evidences are provided due to the complexity of the problems.

In this paper, we propose an algorithm that rigorously proves results on counting central configurations for all masses in one dimensional parameter spaces. Especially, we provide an approach to find all bifurcation points and count real roots at those points, known only implicitly. A spatial restricted \((4+1)\)-body problem and a planar \((1+3)\)-body problem are successfully applied by our method. All results except for the equal masses for the restricted \((4+1)\)-body problem are new and the results for the planar \((1+3)\)-body problem are new at the bifurcation points.

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Acknowledgements

The author would like to thank Professor Richard Moeckel for his Mathematica codes for some of the computations in this paper. This research was partly supported by the Ministry of Science and Technology of the Republic of China under the grant MOST 104-2115-M-005-004.

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Authors and Affiliations

  1. Department of Applied Mathematics, National Chung Hsing University, 145 Xingda Rd., South Dist., Taichung City, 402, Taiwan

    Ya-Lun Tsai

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  1. Ya-Lun Tsai
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Correspondence to Ya-Lun Tsai.

Appendix

Appendix

$$\begin{aligned} g^{4} =& 14348907 k^{8} (-1 + 2 k)^{8} (-209131272894456944593144423783511499+ 12110628365621863141752595158626784528 k\\ &{}- 321833043687522840731801104763386910592 k^{2}+ 5282816045452154761864610890773283600512 k^{3}\\ &{}- 62321870899214934024101752875083881832064 k^{4}+ 599525084805566810091138278319212884859904 k^{5}\\ &{}- 5089568434708560545391461340982194650038272 k^{6}+ 36166744134658030710388619684062815617212416 k^{7}\\ &{}- 173682271713691426508982988802998229113860096 k^{8}+ 194571006705401622461750893770741265329291264 k^{9}\\ &{}+ 4361256499184399605996203588453710463202492416 k^{10}- 36421637749003467158373200672552890309715951616 k^{11}\\ &{}+ 124882780920217517416458592876205800627856474112 k^{12}+ 29686057623670734992158867195772201771747770368 k^{13}\\ &{}- 2466456064067386957557900544447489158933808939008 k^{14}+ 13381022051799846243452346385896117243126868869120 k^{15}\\ &{}- 41374670022892701550349904115844068890901605253120 k^{16}+ 80753323917919406241244407109506118710185454206976 k^{17}\\ &{}- 83736644329390615981846331514643026939784938389504 k^{18}- 30007914497106340313051523250403681924098155347968 k^{19}\\ &{}+ 272731526235208997199716801978606751657411295051776 k^{20}- 474893308685999304513566831791186612587185504780288 k^{21}\\ &{}+ 377936791496790041531539287484101936517848003248128 k^{22}+ 45725571810491945129380762018308178031630427881472 k^{23}\\ &{}- 461651770005669715931369926596333817388529501601792 k^{24}+ 531482425572495898790826588549220379127061961244672 k^{25}\\ &{}- 303978834654331499546743054391553714615507518226432 k^{26}+ 69287711932141399434591769259304063291764502429696 k^{27}\\ &{}+ 27556767991099841456412079571314810360028677013504 k^{28}- 36931239344498259899134718751866345172519622803456 k^{29}\\ &{}+ 25570057182062646531889634545006726676623579414528 k^{30}- 13171219513113041503585322296508115743622003949568 k^{31}\\ &{}+ 4161404052216203236353315247932243607000751013888 k^{32}- 187806263065237241933283988669334605412687675392 k^{33}\\ &{}- 519880180214264903494000068173159853087738822656 k^{34}+ 269442833171764114875905064559700228009694855168 k^{35}\\ &{}- 68433198701898044709157961856663742108338749440 k^{36}+ 6459589270291605471908946749983035900473376768 k^{37}\\ &{}+ 2491259031666340925404431775738992043235475456 k^{38}- 975758706756982810145782951238978471618150400 k^{39}\\ &{}+ 97575870675698281014578295123897847161815040 k^{40}).\\ g^{5} =&1000301832637713093336811104632832 k^{14} (-6193005856 + 34523341224 k- 88164313884 k^{2} + 179027640450 k^{3} - 296194349727 k^{4} + 292634913915 k^{5}\\ &{}- 135093148569 k^{6} + 22052144295 k^{7})^{2} (7290000000000 - 437400000000000 k+ 10671484725000000 k^{2} - 91123902855000000 k^{3} + 643286524024321875 k^{4}\\ &{}- 22708959618399581250 k^{5} + 500801355667529080875 k^{6} - 6197850032153476113900 k^{7} + 54882543612603736731564 k^{8}- 421146331287143356032660 k^{9}\\ &{}+ 2954546557428780047159658 k^{10} - 17648630788414956041955708 k^{11} + 83771969920609859108071258 k^{12}- 306975451272810787413804180 k^{13}\\ &{}+ 857391208251519414055109752 k^{14} - 1794120238816087989509616300 k^{15} + 2726191394983375066395449239 k^{16} - 2838574971479705566887774018 k^{17}\\ &{}+ 1728344797213312641067462323 k^{18} - 37350374248580151836867904 k^{19} - 1300031429522116294856107152 k^{20} + 1983441863951450533463609376 k^{21}\\ &{}- 2053166141818806972470118720 k^{22} + 1531280866164945026551289856 k^{23} - 694191759631599704931041472 k^{24}+ 1583068252243304132527104 k^{25}\\ &{}+ 286525215697323461069387520 k^{26} - 258835251578200089068123136 k^{27} + 133514108743669727266861056 k^{28} - 40315836246102348631824384 k^{29}\\ &{}+ 5280932874361714728127488 k^{30}).\\ g^{6} =&(-10 + k)^{2} (-312 + 1896 k + 530 k^{2} + 588 k^{3} + 101 k^{4} + 31 k^{5} + k^{6})^{2} (2963399638413312 + 23142740033323008 k + 72951703384555008 k^{2}\\ &{}+ 111198530932125696 k^{3} + 57853164006793584 k^{4} - 74362983171264576 k^{5} - 150432849004786240 k^{6} - 90543746418355752 k^{7} + 13327925036628816 k^{8}\\ &{}+ 44887125905050176 k^{9} + 20891973423983679 k^{10})^{3} (-8438743501575408 + 2678966190976320 k + 345472406710968960 k^{2} + 400296455761238280 k^{3}\\ &{}- 11384652544065195120 k^{4} + 12123699015371647968 k^{5} + 119990316536124348321 k^{6} - 159500845521348320736 k^{7} - 1771852062223329593868 k^{8}\\ &{}+ 9249906747903520991616 k^{9} - 25608023464518181047260 k^{10} + 53192685061230729315840 k^{11} - 87060482984162666161152 k^{12}\\ &{}+ 105029467560226902200832 k^{13} - 90330266881601149980672 k^{14} + 54441252684647804362752 k^{15} - 3597828247551090484224 k^{16}\\ &{}- 50207178662075239133184 k^{17} + 45101390498475214934016 k^{18} + 2898856085837444554752 k^{19} - 11491835434562035187712 k^{20}\\ &{}+ 544400220773632720896 k^{22}) (130826199714027687546065564006018101914573913451569359994533161147196600096890290176\\ &{}+ 6379334690817350097484339882960120779072556541638429744495331286415681833295983673344 k\\ &{}+ 143343436290907434190236021879697713112605548862658775303662260908808778086044764471296 k^{2}\\ &{}+ 1925971468600100640078529726392319981686684029582601342104648453967347669944533862318080 k^{3}\\ &{}+ 16429902038782606583966068303176355643061275650052560033878736641831321770977372649505024 k^{4}\\ &{}+ 79818441780782785298622655114118418614152880689308033436557543804575226170277636842717184 k^{5}\\ &{}+ 11861556659920223788540462451206599915743797275844984659267609794402598580893349608768000 k^{6}\\ &{}- 3326425877717349404890766114854188474912014861300004392474168258124949363371872228243945472 k^{7}\\ &{}- 29810408973203405650504903643812092463921474518251682329413306945917906711732246082125358848 k^{8}\\ &{}- 137766675196343189652470538532985423408703481491672330048478847419017900194019901136583450624 k^{9}\\ &{}- 221132170868715922837450062532672166778701316102052890154980626384120653744791062068949029056 k^{10}\\ &{}+ 1807663668476843379787247448654704281223072364906507467853970901734633584037194483383449411584 k^{11}\\ &{}+ 17210516363703688109410764981931187566113319520547741849634865036725432133642018859904797599616 k^{12}\\ &{}+ 76453250500788007077602612456964514915210305030229523300641479285800042789091039931251226507392 k^{13}\\ &{}+ 171826074945197597439477944345955886878696447327774201742732925591358550636049577372250627146816 k^{14}\\ &{}- 187799799219729462130480345019905757098329529988354011206691518378527904690248560149070734150912 k^{15}\\ &{}- 3506935540755201516061628765671398979334892397630228474731645666802791305163082628957931630620928 k^{16}\\ &{}- 16685981007862080222261942014908814524889083182754418387932967366610605254027552236687716057516160 k^{17}\\ &{}- 46741539140815536453664256883776051478980280491609165478123328651577225431704752392429668464103648 k^{18}\\ &{}- 61791521559184706472527367904713974509060889517035425682666785763021907402125508683791314157012224 k^{19}\\ &{}+ 131249601858955428013626830456690535318883698534523207989005957351159703116814171185594919788454000 k^{20}\\ &{}+ 1068544560710046898111443181163038729656721629680505566453439569887009100129604384121851588603022464 k^{21}\\ &{}+ 3463079992414460988274354308988673334223003901096743582305383724385402690601787822817732797861546441 k^{22}\\ &{}+ 6419822113434879250282302315521108299434239825846597981042573749810170217477256298287696571335421600 k^{23}\\ &{}+ 1979519903939803893438172353625722285226670911874921540140986628363321908317603740148445491021677176 k^{24}\\ &{}- 34683376930038393458953245095561876552348902973219361538733201261128603393761859204472584541895138972 k^{25}\\ &{}- 154087104607807508146279666033043838233560701359289712190153578642306380570193723790622660606190028364 k^{26}\\ &{}- 427455243182638450672140169260207191723922410340555485500167115355466526596877319824129201909624857024 k^{27}\\ &{}- 923089682323392891666047074140958455517108052382898305706194205345481921257734025646023101855683462418 k^{28}\\ &{}- 1708972993943732388803634206263203439555789464315773328360019952062724292775643178118739178779929839696 k^{29}\\ &{}- 3011990205852923110945308260891356867708147389086770527692074955503309130927194170373066970226410019726 k^{30}\\ &{}- 5678934487344602279227963368062517721546353608634445744656987743819749786276387550890061336279696102364 k^{31}\\ &{}- 11815279314924989958518720012276790070346076895949444644113895058705170425027501349046603253701891281152 k^{32}\\ &{}- 24801774380754131156491823684222603099127511055198039944478179594317357994308496365195196441449729521496 k^{33}\\ &{}- 47168261827276459298454261797663378900778492070840713684305556059277688132995070307373076527936197929299 k^{34}\\ &{}- 75093411932431688760387739140737443441255559997177462428517787642799225677995702550689867686646689044176 k^{35}\\ &{}- 90735490170765155437366044777137870223414013922975351374908498666176348567238357739348175822928477777300 k^{36}\\ &{}- 57849476055832003614944342766728571184345788180763119952952913029256133284610070847957480209575804762128 k^{37}\\ &{}+ 70880080173819969063467196699223343380987603661719313659638637345083086195282278990831134960221594135595 k^{38}\\ &{}+ 329406620455910920455686252427003454970930641454827294708462044642744244351737699877097388732685595848392 k^{39}\\ &{}+ 702602033267609853865934083513335793819190112296049924813989635004024334123484721444302942726842203139792 k^{40}\\ &{}+ 1099355110516326873784820219647587814489461157592058650684255106276781585915891508539939751252834522838396 k^{41}\\ &{}+ 1360267765310901248406493563365479530444515456076185510570653832713615208384957283039844063779330806071530 k^{42}\\ &{}+ 1312679038581120838812399205562631748751311121093903681353731167035087853084980808751544651121280365610480 k^{43}\\ &{}+ 856425206002602118369511459413241543702348442377962504559503612316575451706497918417802644392316745079506 k^{44}\\ &{}+ 36202229725222209897870509764907554989756506518674102892227680007033063422832296752613334065047564720416 k^{45}\\ &{}- 946839103974113378797000648703545303571428530555410279939368539264263076489821725642429989436746737392464 k^{46}\\ &{}- 1802233567414672516966550805472280866222380869926731160045983498527266251669107588719838398980164400794052 k^{47}\\ &{}- 2267969979723143318867523216896052151550388575255975020412037502920019855688847925523445213287333679308040 k^{48}\\ &{}- 2218801712794043701393753065026915691317666044553362193996205842708962228706503765542477961145186583426768 k^{49}\\ &{}- 1709661473064309759056994753023021253296580645808905363468689485278463305033741264062312174280290326456929 k^{50}\\ &{}- 936781256404498758173082528974828953598912621525992091190348088876666258693337608190909410469912078373792 k^{51}\\ &{}- 144425744928496866602683403003905622920055257849593784349483201075750640540286142719812979895725867575656 k^{52}\\ &{}+ 470972067971017621267656262451844221395791326419647036421835335501596126938170576271471792576057499620064 k^{53}\\ &{}+ 816029153634114312705424072613077541028051456221260430734985492367887742789301731465998419154074331962671 k^{54}\\ &{}+ 901398241862998730216566206136105470383762126179825653452080672348867333156706062519647113136518089668240 k^{55}\\ &{}+ 803997923185930692496647021544550619030769736154446720922427781266229831154361437656218405394908377769368 k^{56}\\ &{}+ 618699993454747560183153212395875363653989069105303181572673005133501803388817511305201261981862448965788 k^{57}\\ &{}+ 422452078464695555474554030742372799830333707469014341405339017232288497504444011172744561925087742514496 k^{58}\\ &{}+ 259611314168884006788277847769604371584536041946756686790189045071201455112557236836100998698597973483168 k^{59}\\ &{}+ 144711522136417430545461503606109880498634371338735959612384211753269240781296504523147401998388531985906 k^{60}\\ &{}+ 73479169050130646151453725878188996814286295581744971021938636684228002715931683620190937389017126091120 k^{61}\\ &{}+ 34055509281654745349183438734975572063260564581942378515054237850639368481616365580386334352784313467002 k^{62}\\ &{}+ 14413329056702784347876012650085728773229504993902823506324224926423511343914366808885183512047100312028 k^{63}\\ &{}+ 5565928139449973907459771580680131070217469889200147395135209213229722441637881976739774535781989837936 k^{64}\\ &{}+ 1957325191797168552171033574826019996992995134519519803260725250424720928804594113485493671220288854824 k^{65}\\ &{}+ 624916837615036060334268602937054130436711615253594348218854326481988693802745891893972282030387564027 k^{66}\\ &{}+ 180383969458627961202989383265868774365499162274931933515735551185358671783002693213520904214192018800 k^{67}\\ &{}+ 46817949116451900797748670248298961704611767473584313537835991741717254695275449730282109053029547852 k^{68}\\ &{}+ 10849875318994151603032797323948467554024140287730197036749234364809086774742896450246169431629221552 k^{69}\\ &{}+ 2225221508950651459592654266022542551525792485857827235132999395189877944485800821244395576606621885 k^{70}\\ &{}+ 399333836316577577906961899511206835549858229188976348726535631026857736989583546786091985945900680 k^{71}\\ &{}+ 61796521467123963355105984078485972861372383293338624242889260861139163023969503929397554019685408 k^{72}\\ &{}+ 8089022711223645600529464708861676042153179708190779666179924655790669763003713977485678758757892 k^{73}\\ &{}+ 872542331281041329790157302889074619752457775630885310148043768798640392163509372031654114995970 k^{74}\\ &{}+ 74745328394994581611779939844485006064910130668357818169305194856380178617038980552481143084848 k^{75}\\ &{}+ 4811350227207088047657661135034745075604151309575889046968705504827542728514813286504294378254 k^{76}\\ &{}+ 212942275657411544202473275950082965976660220469767880277522625858544001198392584154640557888 k^{77}\\ &{}+ 5581022912242884171446108084587147924257178666593492361474466389316172301566873799443094596 k^{78}\\ &{}+ 76335623138985695233552036225597533902650187236945887492566357639490967096736389365202500 k^{79}\\ &{}+ 1550152988085496822992578379085646470777530030706561536010310454161996471356618527375000 k^{80}\\ &{}+ 6912431250445102740059121031244811763320553535207182169998113383153814101007000000000 k^{81}\\ &{}+ 119088766094523880809538001718084393280753588166996644685873362375280914043759765625 k^{82})^{2}. \end{aligned}$$

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Tsai, YL. Counting Central Configurations at the Bifurcation Points. Acta Appl Math 144, 99–120 (2016). https://doi.org/10.1007/s10440-016-0042-9

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